Error message when solving simultaneous differential equations

Question

I am trying to solve the following set of differential equations to obtain $phi[t]$, $y[t]$, and $z[t]$. The 3 differential equations are here:
m y''[t] == -k z[t]^(3/2) - c z'[t] - m g,

φ''[t] == ((-k z[t]^(3/2) - c z'[t]) l Sin[φ[t]] + μ (-k z[t]^(3/2) - c z'[t]) Cos[φ[t]] )/i,

z[t] == y[t] - l Cos[φ[t]]

The constants are defined here:
k = 50000000;
c = 37;
g = 9.81;
μ = 0.3;
i = 10^-5;
l = 0.1;
m = 0.35

The initial conditions are here:
y[0] == 0, y'[0] == -3,
φ[0] == 0.1, φ'[0] == 0.1,
z[0] == 0, z'[0]==0

However, I got the following error message when I tried to use NDSolve:
NDSolve::icfail: Unable to find initial conditions that satisfy the residual function within specified tolerances. Try giving initial conditions for both values and derivatives of the functions.

What exactly do these error messages mean? Is there anything wrong with the differential equations that in fact there are only two unknowns, $y[t]$ and $phi [t]$, and $z[t]$ is redundant? Thank you.
Here's my code:
k = 50000000;
c = 37;
g = 9.81;
[Mu] = 0.3;
i = 10^-5;
l = 0.1;
m = 0.35;
NDSolve[{m y''[t] == -k z[t]^(3/2) - c z'[t] - m g, [CurlyPhi]''[
    t] == ((-k z[t]^(3/2) - c z'[t]) l Sin[[CurlyPhi][
         t]] + [Mu] (-k z[t]^(3/2) - c z'[t]) Cos[[CurlyPhi][t]])/i,
   z[t] == y[t] - l Cos[[CurlyPhi][t]], y[0] == 0, 
  y'[0] == -3, [CurlyPhi][0] == 0, [CurlyPhi]'[0] == 0, z[0] == 0, 
  z'[0] == 0}, {y[t], z[t], [CurlyPhi][t]}, {t, 0, 1}]

morbo · Accepted Answer

This more of an extended comment, but maybe this will help someone get further than me.
First, some starting code:
deqn = {m y''[t] == -k z[t]^(3/2) - c z'[t] - m g, [CurlyPhi]''[t] == ((-k z[t]^(3/2) - c z'[t]) l Sin[[CurlyPhi][t]] + [Mu] (-k z[t]^(3/2) - c z'[t]) Cos[[CurlyPhi][t]])/i,D[z[t] == y[t] - l Cos[[CurlyPhi][t]], {t, 2}]};

params = {k -> 50 10^6, c -> 37, g -> 9.81, [Mu] -> 0.3, i -> 1 10^-5, l -> 0.1, m -> 0.35 } // Rationalize;

Looking at your equations, I see that y[t] shows up in deqn[[1]] as a second derivative and deqn[[2]] isn't coupled directly with y[t]...so I differentiated the third equation twice giving y''[t]...now we can subtract one from the other giving:
sol1 = Solve[deqn[[1]], y''[t]] /. y''[t] -> s1 // First 
sol2 = Solve[deqn[[3]], y''[t]] /. y''[t] -> s2 // First 
sol3 = Solve[s1  - s2  == 0 /. Join[sol1, sol2], z''[t]] // First

Create new deqn2 and ics2:
deqn2 = {deqn[[2]], z''[t] == ((-g m - k z[t]^(3/2) - c Derivative[1][z][t] + l m Cos[[CurlyPhi][t]] Derivative[1][[CurlyPhi]][t]^2 + l m Sin[[CurlyPhi][t]] ([CurlyPhi]^[Prime][Prime])[t])/m};

ics2 = {[CurlyPhi][0] == 0.1, [CurlyPhi]'[0] == 0.1, z[0] == 0, z'[0] == 0};

Solve,
sols = NDSolve[{deqn2 /. params, ics2}, {z[t], [CurlyPhi][t]}, {t, 0,1}] // First

It appears your functions give real and imaginary solutions...
Plot[Re[{z[t], [CurlyPhi][t]} /. sols], {t, 0, 1}, PlotRange -> All]

I admit, I have vague memories of running this kind of method for DAEs of index 1 (which your system appears to be). However. I cannot tell you if this is the actual correct solution to your system or not....But maybe someone smarter than me can take this advice further, Good luck!
update it appears I had changed one of the equations by accident leading it to be solveable in the OPs intended way...I have updated the code with the OPs original code...this has however changed the plot/result, but not the method.

Error message when solving simultaneous differential equations

One Answer

Add your own answers!

Ask a Question